Talk: The complex beauty of fractal geometry

February 22 @ 5:00 pm - 6:00 pm

The first in a series of mathematics and statistics talks aimed at a general audience. This talk is intended to be accessible to secondary students, and will be particularly suitable for A-level students upwards.

No need to book, just turn up!

Geometry is the branch of Mathematics devoted to questions about shapes, relative position and measurement of figures in space, as well as properties of spaces. One of the oldest subfields of Geometry is Euclidean Geometry, a mathematical system attributed to third century BC Greek mathematician Euclid and described in his textbook “The Elements”, which is often referred to as the most successful and influential textbook ever written in history. Indeed, Euclidean Geometry is Geometry in its classical sense: the geometry taught at school, part of the mandatory educational curriculum of the majority of nations in the world. It includes the study of points, lines, triangles, circles, planes, and solid figures. In particular, Euclidean classical geometry deals with objects of integer dimensions: points are zero-dimensional objects; lines and curves are one-dimensional objects, cubes and spheres are three-dimensional objects.

However, by the end of the nineteenth century, mathematicians realised that Euclidean classical objects like points, lines, squares, circles, cones and spheres do not exhaust all possibilities: they discovered shapes that could not be described in terms of simple Euclidean objects. They also realised the converse: that if we want to describe in detail the features of many real objects – such as clouds or mountains or parts of living organisms or lightning in the sky – then we need a different kind of Geometry: Fractal Geometry. As one of the fathers of Fractal Geometry, French-American mathematician Benoit Mandelbrot, used to say, “Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.”

In this lecture, I will explain, with the aid of some computer graphics, how Fractals – the shapes that come out of Fractal Geometry – can be defined as geometrical objects characterised by two properties: self-similarity, and non-integer dimension. Differently from the ‘smooth’ figures of classical Geometry, such as circles or triangles, Fractals turn out to be ‘rough’ and infinitely complex. I will show many examples and images, illustrating how rich and surprisingly beautiful Fractals can be, as well as how intriguing and counter-intuitive Fractal Geometry is and what profound impact it has had in Science and Art.